A333322 - OEIS

A333322

Decimal expansion of (3/8) * sqrt(3).

6, 4, 9, 5, 1, 9, 0, 5, 2, 8, 3, 8, 3, 2, 8, 9, 8, 5, 0, 7, 2, 7, 9, 2, 3, 7, 8, 0, 6, 4, 7, 0, 2, 1, 3, 7, 6, 0, 3, 5, 5, 1, 9, 7, 0, 1, 7, 8, 8, 9, 2, 7, 3, 5, 5, 2, 0, 9, 2, 7, 6, 1, 7, 2, 9, 4, 4, 7, 4, 8, 8, 1, 3, 4, 0, 8, 0, 0, 0, 1, 3, 9, 0, 5, 4, 2, 9, 8, 2, 0, 0, 3, 3, 9, 6, 8, 2, 1, 5, 8, 7, 8, 3, 5, 9, 8, 0, 3, 0, 3, 0, 7, 7, 7, 5, 1, 3, 6, 3, 6

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OFFSET

0,1

COMMENTS

This is the area of the regular hexagon of diameter 1.

From Bernard Schott, Apr 09 2022 and Oct 01 2022: (Start)

For any triangle ABC, where (A,B,C) are the angles:

sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference],

cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference],

and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]:

(ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3).

The equalities are obtained only when triangle ABC is equilateral. (End)

REFERENCES

O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.15, p. 526.

D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111.

LINKS

Table of n, a(n) for n=0..119.

Scott Brown, Problem 3453, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343.

FORMULA

Equals A104954/2 or A104956/4.

EXAMPLE

0.649519052838328985...

MATHEMATICA

RealDigits[(3/8) * Sqrt[3], 10, 120][[1]]

PROG

(PARI) sqrt(27)/8 \\ Charles R Greathouse IV, Apr 09 2022

CROSSREFS